Nested intervals, uncountable sets, unique points.

[mathkiddi]

Homework Statement

Let [a,b] be an interval and let A be a subset of [a,b] and suppose that A is an infinite set.

Suppose that A is uncountable. Prove that there exists a point z which is an element of [a,b] such that A intersect I is uncountable for every open interval I that contains z.

Homework Equations

I don't really know how to start this problem. I know I can use the fact that a set that contains an uncountable subset is uncountable. Any help would be appreciated

The Attempt at a Solution

I know z is an element of I, and that I is uncountable. I also know z is an element of [a_n, b_n]. I know there exists a unique point z in [a_n, b_n] and I know A is uncountable.

 

[matt grime]

Do you know about compactness?

 

[mathkiddi]

no, we have not covered compactness.

 

[Billy Bob]

mathkiddi, what "completeness" axioms/theorems have you covered? Specifically, have you covered any of these:

least upper bound, bounded monotone sequences, Bolzano-Weierstrass, Heine-Borel, nested intervals, or anything like these? limit points? accumulation points? subsequences?

I also know z is an element of [an, bn].

What is [an, bn]?